Most approaches to causal discovery assume a fixed (or time-invariant) causal model; however, in practical situations, especially in neuroscience and economics, causal relations might be time-dependent for various reasons. This paper aims to identify the time-dependent causal relations from observational data. We consider general formulations for time-varying causal modeling on stochastic processes, which can also capture the causal influence from a certain type of unobserved confounders. We focus on two issues: one is whether such a causal model, including the causal direction, is identifiable from observational data; the other is how to estimate such a model in a principled way. We show that under appropriate assumptions, the causal structure is identifiable according to our formulated model. We then propose a principled way for its estimation by extending Gaussian Process regression, which enables an automatic way to learn how the causal model changes over time. Experimental results on both artificial and real data demonstrate the practical usefulness of time-dependent causal modeling and the effectiveness of the proposed approach for estimation.